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Find the equation of the line perpendicular to \(\displaystyle y = \frac{4 x}{3} + 6\) that contains \(\displaystyle \left( -7, \ -8\right)\).

The slope of the line is \(\displaystyle m = \frac{4}{3}\) so the perpendicular line has slope \(\displaystyle m=- \frac{3}{4}\). The equation has the form \(\displaystyle y=- \frac{3}{4}x+b\). Using the point \(\displaystyle \left( -7, \ -8\right)\) gives the equation \(\displaystyle -8=- \frac{3}{4}\left(-7\right)+b\) Solving for \(\displaystyle b\) gives \(\displaystyle b = - \frac{53}{4}\). The equation of the perpendicular line is \(\displaystyle y = - \frac{3 x}{4} - \frac{53}{4}\).

Download \(\LaTeX\) 

\begin{question}Find the equation of the line perpendicular to $y = \frac{4 x}{3} + 6$ that contains $\left( -7, \  -8\right)$. 
    \soln{9cm}{The slope of the line is $m = \frac{4}{3}$ so the perpendicular line has slope $m=- \frac{3}{4}$. The equation has the form $y=- \frac{3}{4}x+b$. Using the point $\left( -7, \  -8\right)$ gives the equation $-8=- \frac{3}{4}\left(-7\right)+b$ Solving for $b$ gives $b = - \frac{53}{4}$.  The equation of the perpendicular line is $y = - \frac{3 x}{4} - \frac{53}{4}$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Find the equation of the line perpendicular to  <img class="equation_image" title=" \displaystyle y = \frac{4 x}{3} + 6 " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%5Cfrac%7B4%20x%7D%7B3%7D%20%2B%206%20" alt="LaTeX:  \displaystyle y = \frac{4 x}{3} + 6 " data-equation-content=" \displaystyle y = \frac{4 x}{3} + 6 " />  that contains  <img class="equation_image" title=" \displaystyle \left( -7, \  -8\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%20-7%2C%20%5C%20%20-8%5Cright%29%20" alt="LaTeX:  \displaystyle \left( -7, \  -8\right) " data-equation-content=" \displaystyle \left( -7, \  -8\right) " /> . </p> </p>
HTML for Canvas
<p> <p>The slope of the line is  <img class="equation_image" title=" \displaystyle m = \frac{4}{3} " src="/equation_images/%20%5Cdisplaystyle%20m%20%3D%20%5Cfrac%7B4%7D%7B3%7D%20" alt="LaTeX:  \displaystyle m = \frac{4}{3} " data-equation-content=" \displaystyle m = \frac{4}{3} " />  so the perpendicular line has slope  <img class="equation_image" title=" \displaystyle m=- \frac{3}{4} " src="/equation_images/%20%5Cdisplaystyle%20m%3D-%20%5Cfrac%7B3%7D%7B4%7D%20" alt="LaTeX:  \displaystyle m=- \frac{3}{4} " data-equation-content=" \displaystyle m=- \frac{3}{4} " /> . The equation has the form  <img class="equation_image" title=" \displaystyle y=- \frac{3}{4}x+b " src="/equation_images/%20%5Cdisplaystyle%20y%3D-%20%5Cfrac%7B3%7D%7B4%7Dx%2Bb%20" alt="LaTeX:  \displaystyle y=- \frac{3}{4}x+b " data-equation-content=" \displaystyle y=- \frac{3}{4}x+b " /> . Using the point  <img class="equation_image" title=" \displaystyle \left( -7, \  -8\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%20-7%2C%20%5C%20%20-8%5Cright%29%20" alt="LaTeX:  \displaystyle \left( -7, \  -8\right) " data-equation-content=" \displaystyle \left( -7, \  -8\right) " />  gives the equation  <img class="equation_image" title=" \displaystyle -8=- \frac{3}{4}\left(-7\right)+b " src="/equation_images/%20%5Cdisplaystyle%20-8%3D-%20%5Cfrac%7B3%7D%7B4%7D%5Cleft%28-7%5Cright%29%2Bb%20" alt="LaTeX:  \displaystyle -8=- \frac{3}{4}\left(-7\right)+b " data-equation-content=" \displaystyle -8=- \frac{3}{4}\left(-7\right)+b " />  Solving for  <img class="equation_image" title=" \displaystyle b " src="/equation_images/%20%5Cdisplaystyle%20b%20" alt="LaTeX:  \displaystyle b " data-equation-content=" \displaystyle b " />  gives  <img class="equation_image" title=" \displaystyle b = - \frac{53}{4} " src="/equation_images/%20%5Cdisplaystyle%20b%20%3D%20-%20%5Cfrac%7B53%7D%7B4%7D%20" alt="LaTeX:  \displaystyle b = - \frac{53}{4} " data-equation-content=" \displaystyle b = - \frac{53}{4} " /> .  The equation of the perpendicular line is  <img class="equation_image" title=" \displaystyle y = - \frac{3 x}{4} - \frac{53}{4} " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20-%20%5Cfrac%7B3%20x%7D%7B4%7D%20-%20%5Cfrac%7B53%7D%7B4%7D%20" alt="LaTeX:  \displaystyle y = - \frac{3 x}{4} - \frac{53}{4} " data-equation-content=" \displaystyle y = - \frac{3 x}{4} - \frac{53}{4} " /> . </p> </p>